As I understand it, in quantum physics if you make a statement like "the system must be either in state A or B at time T, we just don't know which one", the demonstration that you're wrong can only happen in a situation where you measure the system at some
later time T', and the result is such that you retroactively can't be sure whether it was in A or B at the earlier time T. In that case, the rules of probability would say that if really was in state A or B at time T, then the total probability of finding it in state C when you measure it at the later time T' would be given by:
P(system in state C at time T') = P(system in state C at time T',
given that it was in state A at time T)*P(system in state A at time T) + P(system in state C at time T',
given that it was in state B at time T)*P(system in state B at time T)
Or to put it in simpler probability notation:
P(C) = P(C|A)*P(A) + P(C|B)*P(B)
But in certain QM experiments, if you do it this way you get the wrong answer for the probability of getting measurement result C when you measure at T' (an example would be the
double-slit experiment, where the procedure above would lead you to expect the pattern on the screen would just be the sum of two single-slit interference patterns, when what you actually get is a different, double-slit interference pattern). So in the Schroedinger's cat thought-experiment, if you really want to see what's wrong with the assumption that the cat is definitely alive or dead at time T but you just don't know which, then you'd have to imagine something like waiting billions of years after T before opening the box, when the system has had enough time to go to maximum entropy so that all evidence of the cat's past history is lost by the time you open it, so you have no way to deduce whether it was alive or dead at time T. Then if you performed a sufficiently precise measurement of the quantum state of all the particles when you opened the box, the probability of finding them in some particular state S would
not be given by what you'd expect given the assumption it was in a definite state at T, namely:
P(system in state S when box opened) = P(system in state S when box opened,
given that the cat was alive at time T)*P(cat was alive at time T) + P(system in state S when box opened,
given that the cat was dead at time T)*P(cat was dead at time T)
This doesn't necessarily prove that the cat
wasn't definitely alive or dead at time T (in some 'hidden-variable' interpretations like
Bohmian mechanics, it would definitely be one or the other), but if it
was in a definite state, this implies that the probability the system will go to S at a much later time given that the cat was alive at T actually depends on whether or not the cat was
measured by an external observer at time T or not.